Global dynamics analysis of a water hyacinth fish ecological system under impulsive control

Abstract

Control of a water hyacinth-fish ecological system is required for a healthy and sustainable environment. This paper aims to investigate the global dynamics of a water hyacinth fish ecological system under ratio-dependent state impulsive control. First, we study the positivity and boundedness of the solution of the controlled system. By studying the local stability of the equilibrium, we find that the system has two situations. One is that there are two equilibria, namely a saddle point and a boundary equilibrium. In the second case, there are four equilibria, namely, two saddle points, a boundary equilibrium, and a focus point. For the first case, when we select an appropriate ratio-dependent control threshold, the trajectory will globally converge to the boundary equilibrium. For the second case, when the control line is located below the focus point, by using Poincare mapping method, flip bifurcation theory, and vector field analysis techniques, we find that the solution of the controlled system either globally converges to the boundary equilibrium, order-1 periodic solution, or order-2 periodic solution under certain conditions. When the control line is located above the focus point, the solution of the controlled system either globally converges to the focus point, order-1 or order-2 periodic solution. Finally, we use examples to verify the correctness and validity of the theoretical results.

Introduction

Water hyacinth, also known as Eichhornia crassipes, is a floating aquatic plant originating in the Amazon River Basin of South America, and it belongs to the Hyacinthaceae family [1]. Water hyacinths multiplies rapidly under suitable conditions. Once introduced by many countries, water hyacinths is now widely distributed in various parts of the world, and is listed as one of the world’s top 100 alien invasive species [2], casuing serious problems in Asia, Africa, North America and Papua New Guinea. Water hyacinth chokes wetlands and waterways, and thus has become a major pest of creeks, rivers and dams, killing native wildlife. Control of the water hyacinth ecological system is required for a healthy and sustainable environment.

Many environmental threats may arise from the rapid growth of water hyacinth [3]. If not controlled, water hyacinth would rapdily cover lakes and ponds entirely, which can dramatically affect water flow and prevent sunlight from reaching native aquatic plants (often leading to the death of the plants). In addtion, the decay processes depletes the dissolved oxygen in the water and thus can kill fishes and turtles. On the other hand, water hyacinth can also provide a good food source for fishes and turtles. Thus an effective control program in place plays a vital role in establishing a local healthy ecological system in lakes or ponds.

Three commonly used methods to control the growth of water hyacinth are physical, chemical, and biological controls. Each has its own advantages and the optimal control depends on the specific conditions of the affected location such as the extent of water hyacinth infestation, regional climate, and proximity to human and wildlife. Chemical control is the least used out of the three control methods due to its long-term negative effects on the environment and human health. Physical control is performed by land-based machines and mechanical removal is seen as the best short-term solution to the proliferation of the plant. However, it is a costly continual process. As chemical and mechanical removal methods are often polluting, expensive and ineffective, researchers have turned to biological control methods to limit water hyacinth proliferation and suppress its growth.

Limited resources bring many challenges to the control of excessive reproduction and growth of this alien species. How to implement the best control measure to cope with the limited resources has become an important technical issue. The impulsive control strategy is one of the control methods which has been widely used in predator-prey bio-competitive systems. Under impulsive control, the biological and ecological systems can be described by impulsive differential equations, for example, impulsive biological models [4], [5], [6], [7], [8], [9], [10], [11], [13], [14], and other models with discontinuous control [15], [16], [17], [18], [20], [21], [22], [23], [24], [27]. The properties of the solutions of these state impulsive control models, including existence, uniqueness, stability, boundedness, and periodicity, are the main concerns of impulsive differential equations [4], [5], [6], [7], [14], [15], [20], [21]. For instance, Liu et al. [15] investigated a state feedback impulsive model with the transmission and treatment of animal epidemics and proved the existence of order-1 periodic solution. Zhang et al. [4] proposed an impulsive state feedback control strategy for pest management Gompertz model. Fang et al. [16] obtained three order-1 periodic orbits and heteroclinic bifurcation. Fu and Chen [7] studied a water hyacinth ecological system with two state-dependent impulsive controls, and obtained the sufficient condition under which the system has an order-1 or order-2 periodic solution.

State-dependent impulsive control has been widely used to address some real world problems. For example, Zhang et al. [14] discussed the spreading and control of the computer worm and virus of a state feedback impulsive model. Tian et al. [5] considered a classic Lotka-Volterra system with nonlinear state-dependent feedback control, and proved the existence and global stability of an order-1 periodic solution. They also discussed the existence of periodic solutions with an order of 2 or 3. Wang et al. [6] established a predator-prey model for pest control with multi-state dependent impulsive control and studied the stability of the order-1 periodic solution. In addition, some impulsive semi-dynamical system have been reported in the literature. Tian et al. [13] performed the stability analysis of the periodic solution in a semi-continuous dynamic system. Xie and Wang [33] studied the SIQS epidemic model with non-linear incidence and ratio-dependent impulsive control. Nie et al. [27] investigated the SIVS epidemic model with state-dependent pulse vaccination. Impulsive control methods have also been reported in Refs. [17], [18], [19]. for controlling neural networks.

In a water hyacinth-fish ecological system, there is a competitive relationship between water hyacinths and fishes. When the number of water hyacinths reaches a certain value, the state impulsive control can be used to reduce the population of water hyacinths. However, in some cases, when the growth rate of fishes is greater than that of water hyacinth, the control of a single state variable on the number of water hyacinths is not a rational harvesting policy for the ecological system. Instead, control of two state variables would be economically viable and realistic. Therefore, in this paper, we use a ratio-dependent state impulsive control to establish a health water hyacinth-fish ecological system. When the ratio of the populations of water hyacinths to fishes reaches a certain value, the control is implemented. This control strategy can better prevent the overgrowth of individual species.

Motivated by the above discussions, the main purpose of this paper is to perform the global dynamics analysis of a water hyacinth fish ecological system under ratio-dependent state impulsive control strategy. Currently, many discontinuous control strategies have been developed, including the feedback control strategy [4], [9], [11], [16], [21], the sliding mode control strategy [12], and the intelligent control strategy [32]. Nevertheless, in this paper, we design a ratio-dependent state impulsive control strategy for a water hyacinth-fish ecological system.

The main difficulties of this research work are reflected in three aspects. (1) In order to ensure the ecological significance, how to ensure the positiveness of and boundedness of the solution of system Eq. (2.1) under the radio-dependent impulsive control from positive initial values? We will use comparison theorem and combine with the feedback properties of state impulsive control to analysis the solution properties. (2) Since the analytical solution of the controlled system can not be realistically expressed in terms of time and coefficients, how to obtain the existence of periodic solution of the water hyacinth-fish ecological system under the radio-dependent impulsive control? Based on the horizontal asymptote line of system Eq. (2.1) and the characteristic equation line, we will use vector field analysis, a continuous Poincare map and the fixed point theorem to analyze the existence of the periodic solution. (3) After the existence of periodic solutions is confirmed, how to obtain different forms of periodic solutions (such as order-1 and order-2 periodic solutions) for the system under ratio-dependent state impulsive control? As the Floquet multiplier expression is a transcendental function, it is difficult to solve the expression of the solution for a general system. We will utilize the flip bifurcation theorem, comparison theorem and a continuous Poincare map to investigate order-1 and order-2 periodic solutions.

The main contributions of our manuscript include three points:

• Different from the single variable control methods in existing literature [4], [5], [7], [9], [10], we propose a ratio-dependent impulsive control strategy for a water hyacinth fish system. If the water hyacinth fish system with a single variable control strategy, then the uncontrolled variables may increase unrestrictedly or reduce to zero under unrestricted conditions, which is not conducive to the balance and sustainable development of the whole water hyacinth-fish system. The strategy of ratio-dependent state impulsive control can effectively solve this problem.

• Notice that the Floquet multiplier expression is a transcendental function in Refs. [7], [9], [10], it is difficult to solve the expression of its solution in a general case. This leads to some difficulties proving the existence of periodic solutions of the system. In this paper, when the control line is located below the focus point, by using the fixed point theorem, the comparison theorem, and a continuous Poincare map, we find that the system has a globally asymptotically stable boundary equilibrium point under an appropriate critical value. Moreover, the conditions for the existence of orbitally asymptotically stable order-1 or order-2 periodic solutions are obtained by employing the comparison method and using the fixed-point principle.

• In Refs. [7], [9], [10], [31], the single variable impulsive control system was found to have an order-1 or order-2 periodic solution by using the analogue of Poincares criterion. However, when the control line is located above the focus point, the dynamics of system Eq. (2.1) shows some different results by using the method of inequality, Poincare mapping and flip bifurcation. The solution of the system either globally converges to the equilibrium point (a stable node or a stable focus point), the orbitally asymptotically stable order-1 or order-2 periodic solution.

In establishing the impulsive control threshold, many existing studies used only a single variable of prey or predator to introduce a certain value to carry out the impulsive control. Unlike changing the single variable in impulsive control in [7], we will use a certain ratio of prey to predator to trigger the impulsive control for a water hyacinth-fish ecological system. The dynamical behavior analysis of an ecological system under ratio-dependent state impulsive control is more complex than that of the state impulsive system studied in [7], [11], [16], [27], due to the complexity of the ratio-dependent state impulsive control system. In addition, the obtained results show that the state impulsive system studied in this paper can have rich dynamical behaviors.

The rest of this paper is organized as follows: In Section 2, we describe the differential equations of the model and present the positiveness and boundedness of the solution of the system. Then, we study the sufficient conditions of order-1 and order-2 periodic orbits in Section 3. In Section 4, some numerical simulations are provided. Section 5 gives a brief conclusion.